Subgroups — #27277835

Order 13568.

422 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810375	C₂	1	2	yes	maximal
4	C₂×#16810373	C₂	1	2	yes	maximal
5	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₄×#88322	C₄	1	1	yes	characteristic
9	C₂×C₄×#88322	C₄	1	1	yes	characteristic
10	C₂×C₄×#88322	C₄	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810378	—	2	2	no	—
14	#16810375	—	2	4	no	—
15	#16810373	—	2	4	no	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₄³×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	2	no	—
19	C₂²×#88322	—	2	2	no	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄×#88322	—	2	2	no	—
27	C₄×#88322	—	2	2	no	—
28	C₄×#88322	—	2	2	no	—
29	C₄×#88322	C₈	1	4	yes	complemented
30	C₄×#88322	D₄	1	2	yes	—
31	C₄×#88322	D₄	1	2	yes	—
32	C₄×#88322	Q₈	1	2	yes	—
33	C₄×#88322	Q₈	1	2	yes	—
34	C₂²×C₈×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	D₄	1	2	yes	—
38	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	—	4	4	no	—
49	C₄²×C₅₃	—	2	4	no	—
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	—	2	1	no	—
52	C₄²×C₅₃	—	2	4	no	—
53	C₄²×C₅₃	C₂×C₈	1	2	yes	—
54	C₄²×C₅₃	M₁₆	1	2	yes	—
55	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	—	2	1	no	—
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	—	2	1	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	4	no	—
65	C₂×#88322	C₄⋊C₄	1	2	yes	—
66	C₂×#88322	C₄⋊C₄	1	2	yes	—
67	C₂×#88322	C₄⋊C₄	1	2	yes	—
68	C₂×#88322	C₄⋊C₄	1	2	yes	—
69	C₂×#88322	C₂×C₈	1	2	yes	—
70	C₂×#88322	C₂×C₈	1	2	yes	—
71	C₂×#88322	M₁₆	1	2	yes	—
72	C₂×#88322	M₁₆	1	2	yes	—
73	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
80	C₂×C₄×C₅₃	#108	1	4	yes	—
81	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#135	1	4	yes	—
92	C₈×C₅₃	—	8	2	no	—
93	#88322	—	2	2	no	—
94	#88322	—	2	2	no	—
95	#88322	—	8	2	no	—
96	#88322	C₄⋊C₈	1	4	yes	—
97	#88322	C₄⋊C₈	1	4	yes	—
98	#8807	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	#384	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₂²×C₅₃	#423	1	1	yes	characteristic
102	C₂²×C₅₃	#427	1	2	yes	—
103	C₂²×C₅₃	C₂×#108	1	2	yes	—
104	C₄×C₅₃	—	2	4	no	—
105	C₄×C₅₃	—	2	4	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₅₃	#423	1	1	yes	characteristic
111	C₄×C₅₃	#423	1	1	yes	characteristic
112	C₄×C₅₃	#495	1	2	yes	—
113	C₄×C₅₃	#499	1	2	yes	—
114	#1831	—	53	1	no	—
115	#1831	D₅₃	1	1	yes	characteristic
116	C₄²×C₈	—	53	1	no	—
117	C₂×#335	—	53	2	no	—
118	C₂×#334	—	53	2	no	—
119	C₂×C₅₃	#1564	1	2	yes	—
120	C₂×C₅₃	#1568	1	2	yes	—
121	C₂×C₅₃	#1842	1	1	yes	characteristic
122	C₂×C₅₃	#1846	1	1	yes	characteristic
123	C₂×C₅₃	#1848	1	1	yes	characteristic
124	C₂×C₄⋊C₈	—	53	2	no	—
125	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
126	C₂×C₄×C₈	—	53	1	no	—
127	C₂×C₄×C₈	—	53	1	no	—
128	C₂×C₄×C₈	—	53	1	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₂×C₄×C₈	—	53	1	no	—
131	C₂×C₄×C₈	—	53	1	no	—
132	C₄³	C₂×D₅₃	1	1	yes	characteristic
133	#336	—	2	2	no	—
134	#336	—	106	2	no	—
135	#335	—	106	4	no	—
136	#334	—	106	4	no	—
137	C₅₃	#8807	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	53	1	no	—
140	C₂²×C₈	—	53	1	no	—
141	C₂²×C₈	—	53	1	no	—
142	C₂²×C₈	—	53	1	no	—
143	C₂²×C₈	—	106	2	no	—
144	C₂×C₄²	Dic₁₀₆	1	2	yes	—
145	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
148	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	106	2	no	—
152	C₄⋊C₈	—	106	2	no	—
153	C₄×C₈	—	53	2	no	—
154	C₄×C₈	—	53	2	no	—
155	C₄×C₈	—	53	2	no	—
156	C₄×C₈	—	53	2	no	—
157	C₄×C₈	—	53	4	no	—
158	C₄×C₈	—	106	2	no	—
159	C₄×C₈	—	106	2	no	—
160	C₄×C₈	—	106	2	no	—
161	C₂²×C₄	#11719919	1	2	yes	—
162	C₂²×C₄	#11719919	1	2	yes	—
163	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	53	2	no	—
170	C₂×C₈	—	53	2	no	—
171	C₂×C₈	—	53	2	no	—
172	C₂×C₈	—	53	2	no	—
173	C₂×C₈	—	53	2	no	—
174	C₂×C₈	—	53	2	no	—
175	C₂×C₈	—	53	2	no	—
176	C₂×C₈	—	106	1	no	—
177	C₂×C₈	—	106	1	no	—
178	C₂×C₈	—	106	1	no	—
179	C₂×C₈	—	106	1	no	—
180	C₂×C₈	—	212	2	no	—
181	C₂×C₈	—	212	4	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	4	no	—
186	C₄²	C₈×D₅₃	1	2	yes	—
187	C₄²	#11719912	1	2	yes	—
188	C₄²	D₄×D₅₃	1	1	yes	characteristic
189	C₄²	D₄×D₅₃	1	1	yes	characteristic
190	C₄²	Q₈×D₅₃	1	1	yes	characteristic
191	C₄²	Q₈×D₅₃	1	1	yes	characteristic
192	C₂³	C₂×#11719919	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	#13133146	1	4	yes	—
199	C₂×C₄	#13133152	1	4	yes	—
200	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
201	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
202	C₂×C₄	#13133213	1	1	yes	characteristic
203	C₂×C₄	#13133215	1	1	yes	characteristic
204	C₂×C₄	#13133216	1	1	yes	characteristic
205	C₂×C₄	#13133216	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
207	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
208	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
209	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
210	C₂×C₄	#13133237	1	4	yes	—
211	C₈	—	8	2	no	—
212	C₈	—	53	4	no	—
213	C₈	—	53	4	no	—
214	C₈	—	106	2	no	—
215	C₈	—	106	2	no	—
216	C₈	—	424	2	no	—
217	C₂²	#16810492	1	1	yes	characteristic
218	C₂²	C₂×#13133146	1	1	yes	characteristic
219	C₂²	#16810646	1	1	yes	characteristic
220	C₂²	#16810659	1	2	yes	—
221	C₂²	C₂×#13133152	1	2	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
227	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
228	C₄	#16810645	1	1	yes	characteristic
229	C₄	#16810645	1	1	yes	characteristic
230	C₄	#16811005	1	2	yes	—
231	C₄	#16811022	1	2	yes	—
232	C₂	#20273461	1	2	yes	—
233	C₂	#20273476	1	2	yes	—
234	C₂	#20274866	1	1	yes	characteristic
235	C₂	#20274877	1	1	yes	characteristic
236	C₂	#20274886	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#8807
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
29. C₄×#88322	C₈	semidirect
137. C₅₃	#8807	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20274832	2
C₄²×#88322	2
C₂×#16810375	2
C₂×#16810373	2
C₅₃×#1831	2
#8807	53